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We study the Gibbs statistics of high-density hard-core configurations on aunit square lattice $\mathbb{Z}^2$, for a general Euclidean exclusion distance$D$. As a by-product, we solve the disk-packing problem on $\mathbb{Z}^2$ fordisks of diameter $D$. The key point is an analysis of solutions to normequations in $\mathbb{Z}[{\sqrt[6]{-1}}]$. We describe the ground states interms of M-triangles, i.e., non-obtuse $\mathbb{Z}^2$-triangles of a minimalarea with the side-lengths $\geq D$.  There is a finite class (Class S) formed by values $D^2$ generating sliding,a phenomenon leading to countable families of periodic ground states. Weidentify all $D^2$ with sliding. Each of the remaining classes is proven to beinfinite; they are characterized by uniqueness or non-uniqueness of a minimaltriangle for a given $D^2$, up to $\mathbb{Z}^2$-congruencies. For values of$D^2$ with uniqueness (Class A) we describe the periodic ground states asadmissible sub-lattices in $\mathbb{Z}^2$ of maximum density. By using thePirogov-Sinai theory, it allows us to identify the extreme Gibbs measures (purephases) for large values of fugacity and describe symmetries between them.  Next, we analyze the values $D^2$ with non-uniqueness. For some $D^2$ allM-triangles are ${\mathbb{R}}^2$-congruent but not $\mathbb{Z}^2$-congruent(Class B0). For other values of $D^2$ there existnon-${\mathbb{R}}^2$-congruent M-triangles, with different collections ofside-lengths (Class B1). Moreover, there are values $D^2$ for which both casesoccur (Class B2). The large-fugacity phase diagram for Classes B0, B1, B2 isdetermined by dominant ground states.  Classes A, B0-B2 are described in terms of cosets in$\mathbb{Z}[{\sqrt[6]{-1}}]$ by the group of units.

High-density hard-core model on $\mathbb{Z}^2$ and norm equations in ring $\mathbb{Z} [{\sqrt[6]{-1}}]$